Abstract
Combinatorics In this paper, we study (1 : b) Avoider-Enforcer games played on the edge set of the complete graph on n vertices. For every constant k≥3 we analyse the k-star game, where Avoider tries to avoid claiming k edges incident to the same vertex. We consider both versions of Avoider-Enforcer games — the strict and the monotone — and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases fmonF, f-F and f+F, where F is the hypergraph of the game.
Highlights
Let a and b be two positive integers, let X be a finite set and let F ⊆ 2X be a family of subsets of X
It is very natural to play both Avoider-Enforcer and Maker-Breaker games on the edge set of a given graph G, and for G = Kn, the complete graph on n vertices
Our main objective in this paper is to study monotone and strict H-games played on the edges of Kn, where H is the k-star K1,k, denoted by Sk, for any fixed k ≥ 3
Summary
Let a and b be two positive integers, let X be a finite set and let F ⊆ 2X be a family of subsets of X. It is very natural to play both Avoider-Enforcer and Maker-Breaker games on the edge set of a given graph G, and for G = Kn, the complete graph on n vertices. They conjectured that for any fixed graph H, the thresholds fK−H and fK+H are not of the same order of magnitude, and wondered about the connection between monotone H-games and strict H−-games, where H− is H with one edge missing They investigated H-games where H = K3 (a triangle) and H = P3 (a path on three vertices) and established the following: fKmPo3n =. Studying the star game is very natural, since avoiding a k-star in Avoider’s graph is exactly keeping its maximal degree strictly below k We analyse this game, provide explicit winning strategies for both players under both sets of rules, and obtain the following.
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